Problem: All of the 4th grade teachers and students from Loyola went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$4.00$ each for students, and the group paid $$50.00$ in total. The next month, the same group visited a science museum where the tickets cost $$13.00$ each for teachers and $$7.50$ each for students, and the group paid $$97.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+4y = 50}$ ${13x+7.5y = 97}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-13x-8y = -100}$ ${13x+7.5y = 97}$ Add the top and bottom equations together. $ -0.5y = -3 $ $ y = \dfrac{-3}{-0.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {6.5x+4y = 50}$ to find $x$ ${6.5x + 4}{(6)}{= 50}$ $6.5x+24 = 50$ $6.5x = 26$ $x = \dfrac{26}{6.5}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {13x+7.5y = 97}$ and get the same answer for $x$ ${13x + 7.5}{(6)}{= 97}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.